Paint With Image Projection in Texture Paint Mode
While using a Brush Texture you are painting with the source texture-color color-multiplyed by the Brush color (purple in the case below).



Just set it to Full White (RGB=1,1,1), raise the strenght to 1 and you'll be able to paint the exact colors stored in the texture.





If you need mix the colours of the texture with other textures, you can use the node editor: